We've been using the plugin successfully on other sites and this is the first time we've run into the error: "BackUpWordPress has detected a problem. wp-cron.php is returning a 404 Not Found response which could mean cron jobs aren't getting fired properly. BackUpWordPress relies on wp-cron to run scheduled back ups"

The wp-cron file is in the main directory, as it should be. So not sure why a 404 error is being returned. We're also able to manually make a backup without any problems.

The issue is that your wp-cron.php is not returning a 200 response when hit with a http request originating from your own server, it could be several things, most of the time it's an issue with the server / site and not with BackUpWordPress.

Some things you can test are.

Are scheduled posts working? (They use wp-cron too).
Are you hosted on Heart Internet? (wp-cron is known not to work with them).
If you click manual backup does it work?
Try adding define( 'ALTERNATE_WP_CRON', true ); to yourwp-config.php`, do automatic backups work?
Is your site private (I.E. is it behind some kind of authentication, maintenance plugin, .htaccess) if so wp-cron won't work until you remove it, if you are and you temporarily remove the authentication, do backups start working?